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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The asymptotic behaviour of certain integral functions


Author: P. C. Fenton
Journal: Trans. Amer. Math. Soc. 242 (1978), 123-140
MSC: Primary 30A64
MathSciNet review: 0486507
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Abstract: Let$ f(z)$ be an integral function satisfying

$\displaystyle {\int_{}^\infty \{\log \,m(r,f)\, - \,\cos \,\pi \rho \,\log \,M(r,f)\} ^ + }\frac{{dr}}{{{r^{\rho + 1}}}}\, < \,\infty $

and

$\displaystyle 0\, < \,\mathop {\lim }\limits_{\overline {r\, \to \infty } } \,\frac{{\log \,M(r,f)}}{{{r^\rho }}}\, < \,\infty $

for some $ \rho :\,0\, < \,\rho \, < \,1$. It is shown that such functions have regular asymptotic behaviour outside a set of circles with centres $ {\zeta _i}$ and radii $ {t_i}$ for which

$\displaystyle \sum\limits_{i = 1}^\infty {\frac{{{t_i}}}{{\left\vert {{\zeta _i}} \right\vert}}} < \infty $

.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0486507-2
PII: S 0002-9947(1978)0486507-2
Article copyright: © Copyright 1978 American Mathematical Society